DATA STRUCTURES AND ALGORITHMS Lecture Notes 10 Prepared by İnanç TAHRALI

2 ROAD MAP GRAPH ALGORITHMS Definition Representation of Graphs Topological Sort

3 GRAPHS A graph is a tuple G=(V,E) V : set of vertices /nodes E : set of edges each edge is a pair (v,w), where v,w Є V If the pairs are ordered, then the graph is directed directed graphs are sometimes refered as digraphs Vertex w is adjacent to v iff (v,w) Є E

4 GRAPHS A path is a sequence of vertices w 1, w 2, …, w N where (w i, w i+1 ) Є E for 1≤i<N The length of a path is the number of edges on the path, which is equal to N-1 if path contains no edges, length is 0 If the graph contains an edge (v,v) from a vertex to itself, then the path (v,v) is sometimes refered to a loop. A simple path is a path such that all vertices are distinct, except that the first and last could be the same

5 GRAPHS A cycle is a path with length ≥ 1 where w 1 = w N Cycle is simple, if the path is simple In an undirected graph edges should be distinct for simple cycle A directed graph is acyclic if it has no cycles A directed acyclic graph is refered to DAG An undirected graph is connected if there is a path between each pair of vertices A directed graph is strongly connected if there is a path between each pair of vertices A directed graph is weakly connected if the underlying undirected graph is connected A complete graph is a graph in which there is an edge between every pair of vertices

6 GRAPHS Example : Airport system Each airport is a vertex Two vertices are connected by an egde if there is a nonstop flight from the airports that are represented by the vertices The edge could have a weight, representing time, distance or cost of flight. Such a graph is directed It might have longer or cost more to fly in different directions Make sure that the airport system is strongly connected It is always possible to fly from any airport to any other

7 GRAPHS Traffic flow can be modeled by a graph Each street intersection represents a vertex, each street is an edge. Edge costs could represent, among other things, a speed limit or capacity We could ask for the shortest route or use this information to find the most likely location for bottleneck

8 Representation of Graphs We will consider directed graphs We number the vertices, starting at 1. The graph below represents 7 vertices and 12 edges

9 Representation of Graphs Adjacency Matrix Representation Use a two dimensional array to represent a graph For each edge (u,v) Є E we set A [u][v] = 1 A [u][v] = 0, otherwise If the edge has a weight we set A[u][v] = weight we can use -∞ / ∞ to indicate nonexistent edges Space requirement = θ(|v| 2 ) An adjacency matrix is an appropriate representation if the graph is dense: |E| = θ (|v| 2 ) is it true in most applications ?

10 Representation of Graphs Adjacency List Representation If the graph is not dense (is sparse) a better solution is adjacency list representation For each vertex, we keep a list of all adjacent vertices. The space requirement is O(|E|+|V|)

11 Representation of Graphs Adjacency list representation of a graph

12 Topological Sort A topological sort is an ordering of vertices in a directed acyclic graph If there is a path v i to v j, v j appears after v i in the ordering Example : course prerequisite structure

13 Topological Sort course prerequisite example

14 Topological Sort If a graph has a cycle, a topological sort is not possible Ordering is not necessarily unique, any legal ordering will do On the example below, v 1, v 2, v 5, v 4, v 3, v 7, v 6 and v 1, v 2, v 5, v 4, v 7, v 3, v 6 are both topological orderings

15 Topological Sort Algorithm Repeat find a vertex with no incoming edges print the node remove it and its edges Until the graph is empty How to find a vertex with no coming edges ?

16 Topological Sort To find a vertex with no coming edges Linear search on vertices O(|V|)for each find O(|V| 2 ) for topological sort Keep a list (queue or stack) for vertices with zero incoming edges Update the list when an edge is deleted O(|E|)  if adjacency list is used. Total time O(|E|+|V|)

17 /*Simple topological sort pseudocode void Graph::topsort() { Vertex v, w; for (int counter = 0; counter < NUM_VERTICES; counter ++) { v = findNewVertexOfDegreeZero (); if ( v==NOT_A_VERTEX ) throw CycleFound (); v.TopNum = counter ; for each w adjacent to v w.indegree-- ; }

18 Result of Applying topological sort to the graph below

19 /* Pseudocode to perform topological sort void Graph::topsort() { Queue q(NUM_VERTICES); int counter = 0; Vertex v, w; q.makeEmpty(); for each vertex v if (v.indegree == 0) q.enqueue(v); while (!q.isEmpty()) { v = q.dequeue(); v.topNum = ++counter; // assign next number for each w adjacent to v if (--w.indegree == 0) q.enqueue (w); } if (counter !=NUM_VERTICES) throw CycleFound (); }